Chapter 3

Synchrotron radiation

The purpose of this chapter is to introduce the phenomenon of synchrotron

radiation, and its place in studies of radio-loud AGN. The derivations pre-

sented in Section 3.2 are not original – they owe much of their form to

Longair (1994) – while those presented in later sections are the author’s

own developments starting from the work in Section 3.2.

First, a note: the derivations that follow are aimed at calculating the

luminosity of synchrotron emission. This should (unless stated otherwise)

be taken to mean the spectral luminosity, or luminosity per unit frequency,

with SI units of W Hz−1.

3.1 Introduction

In the 1940s, the development of experimental particle physics led to the

development of various types of particle accelerators, which were used to

accelerate particles such as electrons to high enough energies to observe

particular interactions. One such form of accelerator that was common

was the synchrotron, which used circular magnets to accelerate particles

up to high energies – an early example was the General Electric 70 MeV

synchrotron.

When these synchrotrons were first constructed and utilised, it was no-

ticed (Elder et al. 1947, 1948) that an intense light (polarised in the plane of

the electrons’ orbit) was emitted from the high energy electrons. The theory

of this radiation was investigated by Schwinger (1949), who first derived the

spectrum of radiation that is emitted.

28 Synchrotron radiation

It was a few years later that its application to astrophysics was realised.

Shklovsky (1953) showed that the radio and at least part of the optical emis-

sion of the Crab Nebula could be explained by synchrotron emission. It soon

became accepted as the dominant component of the radio emission from the

Galaxy, from supernova remnants, and from radio galaxies. The natural

polarisation of synchrotron emission was used in a number of instances to

confirm its presence, particularly at optical wavelengths (for example, Baade

(1956) observed the polarisation of the jet in M87). A good review of the

early studies of synchrotron emission is that by Ginzburg and Syrovatskii

(1965), who, in this and their later review (Ginzburg and Syrovatskii 1969),

tried to coin the term “magnetobremsstrahlung” as an alternative to “syn-

chrotron”. However, they did note that “it hardly seems possible at this

late stage to change the accepted terminology”, and sadly, they proved to be

correct.

3.2 Tangled magnetic fields

3.2.1 Introduction

The first derivation presented will be the derivation most commonly used

in the literature. The method followed is that detailed in Longair (1994),

which has the advantage of being done in S.I. units (unlike other derivations

such as those found in Pacholczyk (1970) or Rybicki and Lightman (1979)).

The physical model for this derivation is a jet that has a tangled magnetic

field. This means that there is no preferred angle to the magnetic field lines,

and so we integrate over all viewing angles. The viewing angle is important

because the radiation from each particle is strongly beamed in the direction

in which that particle is travelling, since the particle is relativistic.

This relativistic motion means that the dipole radiation pattern that

is emitted in the rest frame of the particle is Lorentz-transformed into a

strongly beamed pattern in the observer’s, or lab, frame (see Fig. 3.1). The

way this pattern transforms is governed by the aberration formulae:

sinφ =sinφ′

γ(1 + β cosφ′), cosφ =

cosφ′ + β

1 + β cosφ′(3.1)

where a prime (′) indicates the particle’s rest frame. In the rest frame, there

is zero power emitted at angle φ′ = π/2, and so in the lab frame we have

3.2 Tangled magnetic fields 29

Figure 3.1: Diagram showing the effect of relativistic motion at right angles to accel-eration. a) Dipole radiation pattern from a non-relativistic accelerated charge (i.e. in theparticle’s rest frame). b) Beamed radiation pattern from particle observed to be movingto the right. In this case, the particle is moving at a velocity of 0.4c (i.e. a Lorentz factorof just γ = 1.091), and the scale in both the x and y directions has been shrunk by afactor of two.

sinφ = 1/γ, which, for large γ (i.e. for a more relativistic particle), gives

φ ∼ 1/γ. Thus all the forward power is radiated in a beam of angle 2/γ.The emitted power as a function of angle is also affected by the relativistic

motion of the particle. The full expression is not shown here, but it results

in very strong power emitted in the forward direction, and much less power

in the opposite direction (see Fig. 3.1).

3.2.2 Derivation

We first consider the case of a single particle (of mass m, energy E = γmc2,

and charge e) moving with velocity v in a uniform magnetic field of strength

B. The trajectory of this particle makes an angle of αp with the direction

of the magnetic field – this angle is called the “pitch angle”. The particle’s

trajectory forms a spiral, or helix, centred on the field line, due to the ~v× ~B

force. This spiral has a radius of curvature, with respect to the central line,

of a = v/(ωr sinαp), where ωr = eB/mγ is the relativistic gyro-frequency.

The emission can be expressed in terms of components in two polari-

sations: both perpendicular and parallel to the projected direction of the

magnetic field. A rather complete derivation of these two components is

given in Longair (1994) – here we simply give the final result. Defining two

variables to be used in the equations:

θ2γ = 1 + γ2θ2 η = ωaθ3

γ/3cγ3,

30 Synchrotron radiation

we can write these two emission components as:

dI⊥(ω)

dΩ=

e2ω2

12π3ε0c

(

aθ2γ

cγ2

)2

K 2

3

2(η) (3.2)

dI‖(ω)

dΩ=

e2ω2θ2

12π3ε0c

(

aθγcγ

)2

K 13

2(η) (3.3)

where K 1

3

and K 23

are the modified Bessel functions of order 13 and

23 respec-

tively.

The next step is to integrate over solid angle. Since nearly all the ra-

diation is emitted within very small angles of the pitch angle (due to the

relativistic beaming of the radiation pattern), and the elemental solid angle

varies little over dθ , the solid angle becomes dΩ = 2π sinαpdθ. We can

then integrate Eqns. 3.2 & 3.3 with respect to θ. Also, since the radiation

is concentrated in a small angle about αp, we can take the integral limits to

±∞ (as this makes finding an analytic solution that much easier!). Thus:

I⊥(ω) =e2ω2a2 sinαp

6π2ε0c3γ4

∫ +∞

−∞θ4γK 2

3

2(η)dθ (3.4)

I‖(ω) =e2ω2a2 sinαp

6π2ε0c3γ2

∫ +∞

−∞θ2θ2

γK 13

2(η)dθ (3.5)

[Note that by integrating over solid angle, we are evaluating the total power

radiated by the particle, in all directions, which, while useful for the case of

a tangled field, is not necessarily what is desired when calculating observed

synchrotron spectra from a uniform field. This point will be addressed in

Section 3.3.]

By using analytic integrals (Westfold 1959), these two quantities can be

written in terms of the functions

F (x) = x

∫ ∞

xK 5

3

(z)dz and G(x) = xK 23

(x) (3.6)

(where x = 2ωa/3cγ3 = 2η/θ3γ). We therefore obtain

I⊥(ω) =

√3e2γ sinαp

8πε0c3(F (x) +G(x))

I‖(ω) =

√3e2γ sinαp

8πε0c3(F (x)−G(x))

3.2 Tangled magnetic fields 31

Figure 3.2: Synchrotron spectrum of a single particle in a tangled magnetic field. Valuesof the parameters are: B = 10−4T, γ = 104, and αp = 45.

The next and final step in deriving the single particle emission function

is to add these terms and divide by the time over which the radiation was

emitted. Since we have integrated over solid angle, the above expressions

give the energy emitted during one period of the particle’s orbit, i.e. in a

time Tr = ν−1r = 2πγm/eB. Thus, the total luminosity (i.e. radiated power)

of the particle in both polarisations is:

L(ω) =I⊥ + I‖

Tr=

√3e3B sinαp

8π2ε0cmF (x) (3.7)

This flux is shown in Fig. 3.2, for B = 10−4T, γ = 104, and αp = 45.

3.2.3 Populations

This equation gives the emission spectrum for a single particle in a mag-

netic field. More interesting, from an astronomical point of view, is the

spectrum from a population of particles. Such a population will have a

distribution of energies (and hence Lorentz factors) and pitch angles. The

two most commonly used distributions are a power law distribution of ener-

gies, N(E)dE = κE−pdE, and an isotropic probability distribution of pitch

32 Synchrotron radiation

angles, p(αp) =12 sinαp.

We can then integrate over energy and pitch angle to obtain the total

emission from such a population. This is done in detail by Longair (1994),

but we just quote here the result for the dependence of the resulting flux on

both frequency and magnetic field strength:

Lpop(ν) ∝ B(p+1)/2ν−(p−1)/2 (3.8)

This is a very important result, as it links an observable parameter (i.e. the

spectral index of the emission) to the power law slope of the particle energy

distribution, in the sense that α = (p − 1)/2 (where we use α to representthe spectral index, as is standard in the literature). We will come back to

this result in the next section.

Note that a power law distribution of energies produces a power law

emission. In physical systems, however, the energy distribution can often

either cut off, or change slope at some energy. A good example for this

comes from the modelling done by Meisenheimer et al. (1996) on the radio

to optical emission from the jet of M87. They found that the best fit to the

flux measurements was for an energy distribution of the form N(γ) ∝ γ−2.31,

with an abrupt cut off at some maximum energy.

In general, if the energy distribution has a cut off, then the resulting

emission spectrum will have a critical frequency νc where the power law

spectrum has an exponential cutoff (in the same manner as the single particle

spectrum – see Fig. 3.2). It can be shown (for example, Blandford 1990) that

this critical frequency scales like νc ∝ γ2+B, where γ+ here is the maximum

value of the γ distribution.

3.3 Ordered jets

The derivation in Section 3.2 integrates over all the possible viewing angles.

This is valid if the magnetic field in which the particles are moving is tangled

(i.e. there is no preferred direction of the magnetic field), or if you are inter-

ested in the total energy radiated from a region. In the former case, there

will be particles moving in many different directions, and so the observer

will see contributions from many different values of the viewing angle θ.

However, if the magnetic field is not tangled, but is instead quite uniform,

3.3 Ordered jets 33

the observer will only see emission in a small range of θ values, those close

to the angle between the line of sight and the magnetic field direction. Such

a physical set-up may be similar to that seen in the collimated jets of FR II

radio galaxies, or in the high-polarisation regions of both radio and optical

jets. To find the observed spectrum for this case, we need to take into

account only that radiation which is beamed in the direction of the observer.

The total energy emitted as a function of viewing angle is given by the

sum of Eqns. 3.2 & 3.3. To obtain the observed luminosity (that is, the

emitted power per unit frequency) of a single particle, we divide this by the

orbital time Tr = ν−1r = 2πγm/eB. We use this time since the observer

will see a pulse of emission (corresponding to the beamed radiation pattern)

once in every orbit of the helical path of the emitting particle. This gives

the following:

L1p(ω, θ, γ) =eB

2πγm

e2ω2

12π3ε0c

(

aθγcγ

)2[

θ2γ

γ2K 2

3

2(η) + θ2K 1

3

2(η)

]

=e3Bω2

24π4ε0cmγ

(

mβγ

eB sinαp

)2 θ4γ

γ4

[

K 23

2(η) +θ2γ2

θ2γ

K 13

2(η)

]

=e m ω2

24π4ε0Bc sin2 αp

β2θ4γ

γ3

[

K 23

2(η) +(θ2

γ − 1)θ2γ

K 13

2(η)

]

So, if for simplicity we let ζ = em/(24π4ε0c), then the power per unit

frequency for a single particle, as a function of frequency (ν = ω/2π), angle

(θ = φ−αp, where φ is the line-of-sight angle to the magnetic field direction)

and energy (E = γmc2) is given by:

L1p(ω, θ, γ) =ζ ω2

B sin2 αp

β2θ4γ

γ3

[

K 2

3

2(η) +(θ2

γ − 1)θ2γ

K 1

3

2(η)

]

. (3.9)

3.3.1 Populations

Once again, we want to look at the emission from a population of particles,

rather than just a single particle. We use the distributions of energy and

pitch angle used for the previous case.

34 Synchrotron radiation

Energy

The energies of the particles in the population are assumed to be distributed

according to a power law (here we write it in terms of γ, rather than E). In

this case, however, the power law is assumed to cut off at some maximum

energy γ+:

N(γ) dγ =

κγ−p dγ 1 ≤ γ ≤ γ+,

0 γ > γ+.(3.10)

Pitch Angle

The pitch angles are assumed to have an isotropic distribution. This is given

by the probability distribution

p(αp) dαp =1

2sinαp dαp

In general, the distribution of pitch angles will be defined by (or at least

related to) the process by which the particles are accelerated.

Due to the beaming of the emission, an observer is only going to see emis-

sion from particles with pitch angles very close to αp = φ. That is, the range

of αp values is restricted to a small range αp1−αp2

, where αp1= φ−δαp and

αp2= φ+δαp. In practice, since these integrals are calculated computation-

ally, we can determine the value of δαp empirically by finding where the flux

drops below some cutoff level. We find that δαp is roughly proportional to

the inverse square-root of the frequency (at higher frequencies, the emission

becomes slightly more beamed, and so the width of the integration does not

need to be as great), so that

δαp = 10−3

(

ν

(1010Hz)

)− 1

2

, (3.11)

where δαp is measured in radians.

3.4 Exploring parameter space 35

Total emission

To obtain the total emission from such a population, we integrate over both

the energy and the pitch angle, to find:

Ltot(ω, φ) =

∫ αp2

αp1

1

2sinαp

(∫ γ+

1κγ−pL1p(ω, γ, φ− αp) dγ

)

dαp

=ζ ω2κ

2B

∫ αp2

αp1

∫ γ+

1

β2θ4γγ

−p−3

sinαp

[

K 23

2(η) +(θ2

γ − 1)θ2γ

K 1

3

2(η)

]

dγ dαp,

(3.12)

where θ = φ− αp. See the following section for example spectra.

3.4 Exploring parameter space

Examination of Eqn. 3.12 shows that the luminosity at frequency ω of a

population of particles depends on a number of factors: the viewing angle

φ (that is, the angle the line of sight makes with the direction of the mag-

netic field); the particle energy distribution (governed in this case by the

parameters p and γ+); and the magnetic field strength B in the region of

emission. In this section, we will examine each of these individually, to try

to determine the dependence the luminosity has on each of them.

3.4.1 Variation of viewing angle

The viewing angle, or angle of the ~B vectors to the line of sight, is an

important physical quantity, since jets are long, thin structures and will be

inclined at some angle to the line of sight. See Section 2.2.2 for some typical

values for jets that have been observed at optical wavelengths. Note that in

measuring the viewing angle with respect to the direction of the jet, we are

assuming that the magnetic field lies along the jet (i.e. parallel to the jet’s

direction). While this is obviously going to be a simplification of real jets,

observations of M87 (Perlman et al. 1999) indicate that the magnetic field

does tend to be parallel to the jet direction, particularly in the inter-knot

regions.

In Fig. 3.3, we see the variation of the spectrum with changing viewing

angle φ. Shown are spectra in units of both Lν (luminosity per unit fre-

quency) and Lλ (luminosity per unit wavelength), in arbitrary luminosity

36 Synchrotron radiation

units.

Because the Lλ spectra rise to a peak and then decrease (this peak is

related to the critical frequency νc mentioned earlier), we can use the location

of the peak to find dependencies on φ of both the normalisation of the flux,

and the location of the peak wavelength. We find that both these quantities

depend on sinφ in the following manner:

λpeak ∝ (sinφ)−1 (3.13)

Lλ,peak ∝ (sinφ)2.5 (3.14)

Also, the shape of the spectrum is independent of the viewing angle, so that

the entire curve has the same flux–angle dependence.

3.4.2 Variation of magnetic field strength

Another important physical quantity is the magnetic field strength B. A

strong magnetic field will increase the strength of the synchrotron emission,

as well as affect the location of the turnover. Magnetic fields in jets are

often estimated assuming “equipartition”, where the energy densities of the

electrons and the magnetic field are equal. Typical values obtained are

∼ 10−4G = 10−8T, although it is thought that values less than this are

required to account for observed synchrotron cooling (e.g. in the M87 jet

Heinz and Begelman (1997)). The value of the magnetic field is likely to

increase as you move in towards the compact core.

In Fig. 3.4, we see the variation in the spectra as the magnetic field

strength varies from 10−4T to 10−11T. The peak wavelength varies with B

as

λp ∝ B−1 (3.15)

The peak flux, however, does not show a simple power law variation with

B, but rather shows some curvature over the range of values explored. At

large values of B (B > 10−7T), the flux goes like Lp ∝ B2.5, while at lower

values (B < 10−8T), the relationship is Lp ∝ B3.

3.4.3 Variation of energy distribution

The energy distribution (Eqn. 3.10), being a power law up to a maximum

energy, has two parameters: γ+, the maximum energy, and p, the slope of

3.4

Exploringparameterspace

37

Figure 3.3: Variation of luminosity with changing viewing angle φ, which takes the values 90, 70, 45, 30, 20, 15, 10, 5and 1(with largerφ values giving larger fluxes). The values of the other key parameters are indicated.

38

Synchrotronradiation

Figure 3.4: Variation of synchrotron spectrum with changing magnetic field strength B. The values of B considered are every half-dex in therange 10−4T to 10−11T, with higher B values giving higher fluxes. The values of the other key parameters are indicated.

3.4 Exploring parameter space 39

the power law. We need to investigate what dependence the spectrum has

on both of these parameters.

Maximum energy

The maximum energy of the energy distribution will strongly affect the

resulting synchrotron emission, as the higher energies will contribute at the

high-frequency end. A lack of high energy particles, therefore, will result in

a spectrum that turns over at lower frequencies.

A range of synchrotron spectra with different γ+ values are shown in

Fig. 3.5. Note that, for wavelengths longer than the peak, the power law

slope is the same, but the different γ+ values cause the spectrum to turn over

at different points. Once again, we can look at how the peak wavelength

and flux vary with γ+, and we find (looking at the Lλ spectra):

Lλ,peak ∝ γ3+ (3.16)

λpeak ∝ γ−2+ (3.17)

Slope of power law

The slope of the power law determines how rapidly the energy spectrum

drops off with increasing energy. A large value of p means a steeper power

law slope, which means that the higher energies are less prominent in the

particle population. The value of p will affect the slope of the synchrotron

spectrum at frequencies ν ¿ νc.

The spectra for different values of p are shown in Fig. 3.6, from p = 1 (the

one with the largest luminosity) up to p = 4.5 (the least luminosity). As can

be seen, the synchrotron power law index α gradually steepens as p increases,

and we find that the relationship between the two indices is α = (p− 1)/2,which is the same as that for the tangled field case (Section 3.2).

Also worth noting is the fact that the location of the peak does not

change with the different values of p. This removes one degree of freedom

in choosing the location of the peak/turnover frequency of the synchrotron

emission.

40

Synchrotronradiation

Figure 3.5: Variation of synchrotron spectrum with changing maximum energy γ+. The values of γ+ considered are every half-dex in the range10 to 106, with the higher γ+ values giving spectra extending to higher frequencies. The values of the other key parameters are also indicated.

3.4

Exploringparameterspace

41

Figure 3.6: Variation of synchrotron spectrum with changing p. The values of p are spaced every 0.5 in the range 1 to 4.5, and the larger thevalue of p, the steeper and lower in flux is the resulting spectrum. The values of the other key parameters are also indicated. Note that this graphhas a vertical axis that is different from the other graphs.

42 Synchrotron radiation

3.4.4 Overall variation

If we combine all these variations together, we can find how the peak wave-

length and the peak flux vary with all the physical parameters. The net

dependency is:

λpeak ∝ γ−2+ B−1 sinφ−1 (3.18)

or, in terms of frequency:

νc ∝ γ2+B sinφ

This has the same dependency as the tangled field derivation, although

with the added dependence on the line of sight angle. This angle is also

crucial in determining the overall luminosity, due to the strong dependence

found: L ∝ (sinφ)2.5. This is worth noting carefully, since for small anglesto the line of sight the overall luminosity will decrease quite rapidly. This is

the opposite effect that you get from Doppler boosting, an enhancement of

intensity coming from a source moving relativistically towards the observer.

This effect is discussed in the following section.

3.5 Relativistic Doppler boosting

The calculations up to now have been made in the rest frame of the jet,

i.e. there is no bulk motion present. However, in real jets, relativistic bulk

motion has been postulated to exist, mainly from observations of superlu-

minal motion in radio jets. Various models have been developed (Lind and

Blandford 1985; Urry and Padovani 1995) to explain how the observed emis-

sion from a jet with bulk motion is affected, and we investigate two of the

standard models in this section.

For a relativistic source moving at an angle φ to the line of sight, with

a Lorentz factor Γ = 1/√

1− β2 (where β = v/c), the observed flux is

enhanced by the Doppler factor

δ =1

Γ(1− β cosφ)(3.19)

The degree of enhancement depends on the nature of the source, but, in the

case of a continuous, collimated jet (such as is modelled in Section 3.3), it

takes the form

Fν(ν) = δ2+αF ′ν′(ν),

3.5 Relativistic Doppler boosting 43

where a ′ indicates a quantity is measured in the rest frame, ν = δν ′, and

Fν ∝ ν−α. The other basic model for beaming is the case where, instead of

a continuous jet, the outflow occurs in discrete “blobs”. The beaming then

has the form

Fν(ν) = δ3+αF ′ν′(ν).

Note that here, Γ is the bulk Lorentz factor, which relates to the velocity of

features moving along the jet, not the velocities of the individual particles

(which have Lorentz factors γ). This may be due to some bulk flow of

plasma, or, in the latter case, the velocities of the individual “blobs”.

Note that as φ→ 0, the value of δ increases, reaching a maximum valueat φ = 0 of δmax = 1/Γ(1 − β) ≈ 2Γ for β ≈ 1. The effect of Dopplerboosting is shown in Fig. 3.7, for two extreme cases: δ2.5, corresponding to

a continuous jet with α = 0.5; and δ4, corresponding to a jet made up of

discrete blobs with α = 1.0. The flux in these plots are normalised to the

Γ = 1 curve, corresponding to isotropic emission. Note that for increasing

Γ, the flux at small angles rapidly increases, whereas the large angle flux

decreases below the un-boosted flux (the so-called “de-beaming”). The case

of a larger dependence on δ (the discrete blob case) shows a much larger

dependence on angle.

However, in the case of synchrotron emission from an ordered field, this

dependence is counteracted by the angular dependence of the emission itself.

To find the angular dependence of the observed flux, we need to take into

account the fact that beaming changes the observed angle (in the manner of

the aberration formulae Eqn. 3.1). The flux observed at an angle φ in the

observer’s frame will depend on both the Doppler beaming factor and the

flux emitted at angle φ′ in the rest frame of the blob/jet. These angles are

related by (inverting Eqn. 3.1):

sinφ′ =sinφ

Γ(1− β cosφ)

= δ sinφ

Thus, the observed flux at angle φ is

F (φ) ∝ δp(sinφ′)2.5

∝ δp+2.5(sinφ)2.5

44 Synchrotron radiation

Figure 3.7: The effect of Doppler boosting on isotropic emission, for two different depen-dencies on δ: δ2.5 in the top plot and δ4 in the bottom. The amount of Doppler boostingfor each curve is shown by the value of the bulk Lorentz factor Γ next to the terminatingpoint of the curve. A value of Γ = 1 represents no Doppler boosting.

3.5 Relativistic Doppler boosting 45

Figure 3.8: The effect of Doppler boosting combined with the angular dependence ofthe ordered field synchrotron emission (Section 3.3), for two different dependencies on δ:δ2.5 in the top plot and δ4 in the bottom. The Γ values are the same as for Fig. 3.7. Avalue of Γ = 1 represents no Doppler boosting.

46 Synchrotron radiation

where p is either 2 + α or 3 + α depending on the nature of the source (see

above).

Fig. 3.8 shows the angular dependence of the flux for various amounts

of boosting, parametrised by the Lorentz factor Γ, for the same two cases of

Fig. 3.7. The flux is normalised to the observed flux at an angle of 90with no

boosting (that is, with Γ = 1). As can be seen, when the angular dependence

is considered by itself, the flux rapidly decreases for small φ. As more and

more Doppler boosting is added, the flux at small angles increases, due to

both the intensity enhancement and the angular aberration. The boosted

flux approaches that of the isotropic emission case near the peak of the curve,

but drops rapidly at low angles. Also noticeable is that the de-beaming effect

is more pronounced for this case, so that most of the flux is observed in a

small range of angles. As Γ increases, this angular range gets smaller, and

moves to smaller φ values

3.6 Discussion

There are a number of key points to arise from this analysis. Firstly, consider

the effect that the angular dependence of the synchrotron emission has on

the observed flux. In the case of pure Doppler boosting (Fig. 3.7), the flux

at small angles to the line of sight is boosted by factors of up to several

orders of magnitude (depending on the bulk velocity, the spectral slope and

the physical makeup of the jet). However, when this is combined with the

angular dependence of synchrotron emission from an ordered magnetic field

(Fig. 3.8), the observed flux at small angles decreases rapidly. The angles

at which this occurs become smaller for larger values of the bulk velocity Γ.

The flux at large angles is also reduced more, due to enhanced de-beaming.

Thus, for an ordered jet to be viewed at angles close to its direction of

motion (assuming the magnetic field is parallel to that direction), a reason-

ably large bulk velocity (i.e. large Γ) is required. A jet with a low value of

Γ will still have a significant decrease in flux at low values of φ due to the

strong angular dependence of the emission.

Secondly, we note that jets, particularly those seen in double-lobed radio

sources (i.e. such as those in FR IIs), are seen at large angles to the line of

sight. At large angles, the Doppler beaming becomes, in effect, de-beaming,

since the observed flux is reduced. This observation, then, puts a strong

3.6 Discussion 47

restriction on how large the values of Γ can be in the jet. For large enough

Γ, the de-beaming will be sufficient to reduce the jet flux to levels too small to

be observed. Also, this effect applies for both types of synchrotron emission

(both tangled and ordered fields – in fact it is more pronounced for the

ordered field case), and for all types of beaming (i.e. different dependences

on δ).

Finally, since much of the work presented in this chapter has been con-

cerned with synchrotron emission from an ordered magnetic field, is there

any observational evidence that such emission will be important in astro-

physical contexts? There are two particular observations that point to there

being magnetic fields that are quite highly ordered in the jets of AGN.

The first of these observations is the measurement of high values of

polarisation in both the radio and optical – parts of the M87 jet have

P ∼ 40% − 50%. Such high polarisations imply that the magnetic field ishighly ordered. Further examples are the high optical and NIR polarisations

measured from quasars (P ∼ 20% – see Chapter 6). These measurementsare made by integrating over the entire source, and so imply the existence

of regions of highly ordered magnetic fields.

The second observation is that some jets, particularly those in the pow-

erful FR II radio galaxies, are highly collimated. Collimation of a jet

can be caused by either pressure from the ambient medium, or magneto-

hydrodynamic confinement. While the ambient medium is likely to be im-

portant for the weakly confined jets in FRI radio galaxies, it is unlikely that

it can provide the necessary collimation seen in the thin jets in FR II sources.

This collimation can most likely only be provided by highly structured mag-

netic fields that confine the jet to its thin, well collimated shape. Both this

observation and that of high polarisation point to ordered magnetic fields

being important in the jets of radio-loud AGN.

48 Synchrotron radiation